Space-Time Interval

in the theory of relativity, a quantity that characterizes the relation between the spatial distance and the time interval that separate two events. From the mathematical standpoint it is the “distance” between two events in four-dimensional space-time.

In the special theory of relativity the square of the space-time interval s_AB between two events A and B is

where Δt and Δr are, respectively, the time interval and spatial distance between the events and c is the speed of light in a vacuum. The space-time interval between events remains unchanged when one inertial frame of reference is replaced by another; that is, the interval is invariant with respect to Lorentz transformations. By contrast, the quantities of Δr and Δt depend on the choice of frame of reference.

If , the interval is said to be timelike. In this case there exists a frame of reference in which the events occur at a single point in space (Δr = 0), and s_AB = cΔt—that is, the space-time interval is equal to the product of the time interval between the events in this frame and the speed of light.

If , the interval is said to be spacelike. In this case there exists a frame of reference in which the events occur simultaneously (Δt = 0), and the distance between them is Δr = is_AB, where .

When s_AB = 0, the interval is called a null-interval. In this case the relation Δr = cΔt always holds; that is, the events in any frame of reference can be connected by a light signal (seeRELATIVITY, THEORY OF).

In the general theory of relativity, which considers curved space-time in the presence of gravitation, the space-time interval described above may be used only for infinitesimally distant events (seeGRAVITATION).